Lemma 3 (Theorem 1 in [1]}) 
Assume \(\kappa :[-1,1]\rightarrow \mathbb {R}\)  is \(C^{\infty }\)  and has the following asymptotic expansions around \(\pm 1\)
\(\nonumber \kappa (1-t) &=& p_{+1}(t)+c_{+1}t^{\theta } + o(t^{\theta }),\\\nonumber \kappa (-1+t) &=& p_{-1}(t)+c_{-1}t^{\theta } + o(t^{\theta }),\) 
